Government Interventions in a Dynamic Market with
Adverse Selection
William Fuchsy
Andrzej Skrzypacz
February 15, 2014
Abstract
We study government interventions in a dynamic market with asymmetric information. We show that if the government can only carry out budget-neutral policies,
introducing a short tax-exempt trading window followed by short-lived positive taxes
creates a Pareto improvement in the market. Under a su¢ cient condition on the shape
of the gains from trade and the distribution of asset values, we show that, even when
not requiring budget-neutrality, it is optimal to subsidize trades only at time zero while
imposing prohibitively high taxes afterwards. Subsidies can greatly enhance welfare
but they can also be detrimental if they are provided with delay.
1
Introduction
During times of …nancial distress, such as those experienced in 2008 after the demise of
Lehman Brothers, asset sales are an important source of funds for …nancial institutions
such as banks and insurance companies. Unfortunately, the big gains from trade between
those that are liquidity constrained and those that are not may be di¢ cult to realize due
to asymmetric information. As in the classic Akerlof (1970) market for lemons, if buyers
were to pay the price corresponding to the average quality of the assets in the market, sellers
This paper is closely related to our previous working paper: "Costs and Bene…ts of Dynamic Trading in
a Lemons Market."
y
Fuchs: Haas School of Business, University of California Berkeley (e-mail: wfuchs@haas.berkeley.edu).
Skrzypacz: Graduate School of Business, Stanford University (e-mail: skrz@stanford.edu). We thank Ilan
Kremer, Mikhail Panov, Christine Parlour, Aniko Öry, Brett Green, Marina Halac, Johanna He, Alessandro Pavan, Felipe Varas, Robert Wilson, and participants of seminars and conferences for comments and
suggestions. We also gratefully acknowledge support from the National Science Foundation grant #1260853.
1
holding the best assets might not wish to trade. Realizing this, buyers would then reduce
their o¤ers and end up trading with a small fraction of the sellers or none at all. Absent
government intervention, trade either completely stops or slows down, with prices gradually
rising and over time better and better assets being traded in the market.
The main questions we seek to answer in this paper are if and how should the government
intervene in these situations, even if it has a binding budget constraint. We answer them in
a model of a dynamic competitive market in which liquidity-constrained sellers have private
information about their assets and homogenous, liquidity-abundant buyers compete to buy
those assets.
Several recent papers document how di¤erent …nancial markets had drastic reductions
in volume in the aftermath of Lehman Brothers collapse in 2008. Among others, Heider,
Hoerova & Holthausen (2009) discuss the collapse of the interbank market, McCabe (2010)
the money market funds and Du¢ e (2009) discusses the OTC and repo markets. These
contractions were largely driven by the uncertainty over the counterparty’s ability to meet its
obligations and the disagreement over the value of securities that could be used as collateral.
This was clearly re‡ected in the OTC market where the types of securities acceptable as
collateral signi…cantly changed. Information sensitive securities were largely replaced by
cash. Similarly, the assets under management of money market funds saw a big compositional
change at the time of Lehman’s collapse with a pronounced drop in the amount of assetbacked commercial paper and a large increase of government securities. These recent events
motivate our interest in dynamic markets with asymmetric information and the impact of
government interventions on such markets.
Our …rst and main result (Theorem 1) is that, even if the government can use only a
budget-neutral tax/subsidy policy, introducing a short tax-exempt trading window followed
by short-lived positive taxes creates a Pareto improvement in the market. Absent any intervention, the equilibrium is characterized by a smooth ‡ow of trade where worst assets
are sold …rst and both the quality of traded assets and price gradually increase over time.
By taxing future trades, the government creates more incentives to trade in the early taxexempt period. In particular, holders of higher quality assets that would delay trade absent
the government policy now prefer to trade earlier in order to avoid the taxes or excessive
delay. As the quality of the pool of assets sold early improves, market price increases as well.
Higher prices in turn induce even more trade ultimately doubling the amount of trade and
making all sellers better o¤: those who trade early get better prices, those that trade later
save on delay costs (buyers are indi¤erent since they make zero pro…t in either scenario).
Our second result (Theorem 2) is about optimal, i.e., total-welfare maximizing, budget2
neutral interventions. Under a su¢ cient regularity condition on the shape of the gains from
trade and the distribution of asset values, we show it is optimal to subsidize trades at time
zero while imposing prohibitively high taxes afterwards. Intuitively, the regularity condition
implies that the ratio of the marginal gains from trade to the marginal information rents of
the seller is decreasing in asset quality. Under this condition, the solution to the optimization
problem has a bang-bang property: it is optimal to push as much trade as possible to take
place immediately even if it comes at the expense of excluding all higher types from trade
altogether.1 We extend this result in Corollary 1 where we allow the government to carry
out non-budget neutral interventions. This is important for both normative and positive
considerations. We show that there can be big returns from spending some resources in
bailing out these markets: even if raising a dollar in revenues from another market induces
some deadweight loss, it may still be optimal to bailout these markets. This is best illustrated
by showing that in certain cases even with the best budget-neutral intervention the market
completely unravels (see Section 4.3). By providing an initial subsidy the government is able
to jump-start the market and greatly increase the overall surplus.
Our third result stresses the importance of acting quickly. Via a series of examples we
demonstrate that the timing of the subsidies is crucial. If the government moves slowly and
the subsidy is expected to arrive in the future (either deterministically or stochastically) then
it can actually have a negative e¤ect on welfare. The intuition is that the expectation of
future subsidies delays current trade. So, although there is clearly a bene…t from subsidizing
trades, it is of the essence that the government acts fast. While the ‡exibility of the Federal
Reserve and its ability to act fast was likely crucial in the recent crisis, some of the uncertainty
over future interventions/subsidies may have contributed to the reduction in trade volume
in private markets.
Lastly, we note that the exact form of the intervention is not important, as long as the
information rents collected by the sellers are unchanged (and transaction costs are the same).
A proportional subsidy, as assumed for concreteness in the paper, or a government guarantee on the payo¤ of the assets, as implemented during the crisis with the Public-Private
Investment Program for Legacy Assets, would have equivalent e¤ects. Outright purchases
of assets, as implemented with the TARP program, would also be equivalent, but only if the
program budgeting accounted for the expected future proceeds from the assets purchased by
1
This is related to the result that a durable-good monopolist facing a demand curve with decreasing
marginal revenue, would like to commit to set constant price. Such price induces high valuation buyers to
buy immediately and the rest of the buyers never to trade, that is, under commitment the seller would not
want to use time to screen buyers.
3
the government. For example, a subsidy program with a budget b is equivalent to a purchase
program with a budget b; if the budget in the latter case is on net losses from the program
and not on total purchases.2
Related Literature
Optimal government interventions in similar models have been studied recently by Philippon and Skreta (2012) and Tirole (2012). In these papers, the government o¤ers …nancing
to …rms having an investment opportunity and it is secured by assets that the …rms have
private information about (these are sellers in our model - using an asset as a collateral or
selling it to obtain …nancing are essentially economically equivalent). That round of government …nancing is followed by a static competitive market in which …rms that did not receive
funds from the government can raise funds in a private market. This creates a problem of
"mechanism design with a competitive fringe" as named by Philippon and Skreta (2012): the
government intervention a¤ects the post-intervention equilibrium and vice versa (a feature
shared by our model). In sharp contrast to our results, both papers show that tampering
with the private markets does not improve welfare: see Proposition 2 in Tirole (2012) and
Theorem 2 in Philippon and Skreta (2012). Since the post-intervention market creates endogenous IR constraints for the agents participating in the government program, making the
market less attractive could make it easier for the government to intervene. However, these
two papers argue that this is never a good idea.
The key di¤erence between our model and these two papers that leads to these opposing
results is that Philippon and Skreta (2012) and Tirole (2012) assume a static model of the
private market, while we study a dynamic market. It is best seen in the light of our Theorem
2/Corollary 1: under the regularity condition, it is indeed optimal to have government
subsidy at time zero and all trade happening at time zero, with no additional trades in the
future. Beyond this crucial di¤erence in results and their practical interpretation, our paper
di¤ers from these two papers in terms of the focus on the dynamics of trade and the tradeo¤s
in dynamic interventions.
The bang-bang property of the optimal intervention in Theorem 2 is mathematically related to the …ndings of Samuelson (1984). In a static setting he shows that the optimal
budget-neutral mechanism divides sellers into at most three groups: a group that trades
2
That assumes that the government holding the assets to maturity is as e¢ cient as private buyers holding
the asset. If not, then after purchasing the assets the government should pool them into a portfolio and sell
shares of the portfolio to buyers with liquidity. Since the government can commit to pool all of the assets,
there would be no (additional) adverse selection problem in creation and sales of the portfolio.
4
with probably one, a group that trades with a common intermediate probability and a group
that does not trade. In our dynamic setting this translates respectively to a group that trades
immediately, a group that trades with delay and a group that never trades. Our Theorem
2 contributes to his result by establishing a su¢ cient condition for the optimal mechanism
having trade only at t = 0. Moreover, we show how the optimal direct revelation mechanism can be implemented in a decentralized market with a particular government policy
that induces a unique competitive equilibrium. In addition, when the regularity condition is
satis…ed, we also extend the result by allowing for non budget-neutral interventions.
Two related papers, Heider, Hoerova and Holthausen (2009) and Bolton, Santos and
Scheinkman (2011), combine the problem of adverse selection with one of maturity mismatch.
Although we do not model the maturity mismatch problem explicitly, we believe it had
an important role in the recent crisis and our liquidity- constrained sellers likely are in
that situation because of it. That is, we see the maturity mismatch problem as a possible
micro-foundation of our model of gains from trade and asymmetric information. Heider,
Hoerova & Holthausen (2009) have a 3 period model of the interbank market. Banks in
need of liquidity can use the interbank market to borrow from those with excess liquidity.
Asymmetric information about the quality of the assets in the borrower’s balance sheets
makes lenders afraid of lending to a “lemon”leading to a reduction or complete disappearance
of credit. They discuss some policy interventions but their focus is positive rather than
normative and essentially static.3
On the theoretical side, our paper is also related to literature on dynamic markets with
adverse selection. The closest paper is Janssen and Roy (2002) who study competitive equilibria in a market that opens at a …xed frequency. They show that in equilibrium prices
increase over time and eventually every type trades. They do not ask market design or
policy questions as we do in this paper. Yet, we share with their model the observation
that dynamic trading leads to more and more types trading over time. Camargo and Lester
(2013) …nd the same equilibrium dynamics in a setting with decentralized search rather than
a competitive market (in discrete time, with two types of the seller). While their paper is
focused on characterizing the set of equilibria of the game with no government intervention,
they also show that sunset provisions for subsidies can increase bene…ts of government subsidies because, for reasons similar to what we describe in this paper, expectation of future
subsidies can slow down trade earlier on. Our paper shows other examples of problems of de3
Bolton, Santos and Scheinkman (2011) is a bit further from our work since they focus more on the
ex-ante asset choices and they assume there is no asymmetric information initially but rather that it grows
over time.
5
layed/prolonged interventions and adds to this analysis by characterizing good and optimal
polices. On the more technical side, our competitive-equilibrium setup with a continuum of
types and continuous time allows us to show uniqueness of equilibrium, which makes it easier to interpret comparative statics. For other papers on dynamic signaling/screening with
a competitive market see Noldeke and van Damme (1990), Swinkels (1999), Kremer and
Skrzypacz (2007) and Daley and Green (2012). While we share with these papers an interest
in dynamic markets with asymmetric information, none of these papers studies government
interventions.
While in this paper we study government interventions in terms of taxes and subsidies,
there are other ways the government or market designer can a¤ect trade in equilibrium. For
example, in our related working paper, Fuchs and Skrzypacz (2013), we allow the market
designer to determine the times the market should be open or closed. The market microstructure literature (see Biais, Glosten and Spatt (2005)) has also considered the question of how
di¤erent trading protocols perform in the presence of adverse selection. That literature has
mainly focused on the stock markets where there are potentially many competing sellers,
divisible assets and dispersed information. A di¤erent design question for dynamic markets
with asymmetric information is asked in Hörner and Vieille (2009), Kaya and Liu (2012),
Kim (2012) and Fuchs, Öry and Skrzypacz (2012). These papers ask how information about
past rejected o¤ers a¤ects e¢ ciency of trade. Moreno and Wooders (2012) ask a yet another
design question: they compare decentralized search markets with centralized competitive
markets.
2
The Model
There is a mass of size one of …nancially distressed banks (the sellers). Each seller owns one
unit of an indivisible asset. When the seller holds the asset, it generates for him a revenue
stream with net present value c 2 [0; 1] that is private information of the seller. The seller
types, c; are distributed according to F (c) ; which is common knowledge, atomless and has
a continuous, strictly positive density f (c). We assume that the private information is never
revealed.4
There is a competitive market of potential buyers. Each buyer values the asset at v (c)
4
Most of our results can be extended to a setting in which at some deterministic or random time the
private information becomes public, but the players cannot contract on the realization of this information
(see the working paper Fuchs and Skrzypacz 2013). While the cash‡ows generated by the asset (which are
correlated with c) are realized by the buyer, we assume that the buyer and the seller cannot contract on
their realization, i.e., we assume that the seller cannot o¤er warranty contracts on the assets he sells.
6
which is strictly increasing, twice continuously di¤erentiable, and satis…es v (c) > c for all
c 2 (0; 1) ; v (0)
0; and v (1) = 1 (i.e. no gap on the top).5
Time is t 2 [0; 1] and the market is continuously open. There is also a benevolent gov-
ernment that can intervene by subsidizing or taxing trades proportionally.6 The government
publicly commits to a path of taxes
t
for t 2 [0; 1] before the market opens at t = 0: If at
time t buyers pay price pt ; the sellers receive pt (1
t) ;
t
< 0 represents a subsidy.
All players discount payo¤s at a rate r: If bank with type c sells at time t at a price pt ; its
payo¤ is
1
e
rt
c+e
rt
pt (1
t)
and the buyer’s payo¤ at the time of purchase is:
v (c)
pt
Given a path of prices and taxes, the sellers face an optimal stopping problem. Namely,
when to sell and collect pt (1
t)
max
x
Z
:
x
e
rt
rcdt + e
rx
px (1
(1)
x) :
0
Since the stopping problem is supermodular in c and x; if seller of type c has an optimal
stopping time t then all types c0 < c have optimal stopping times t0
t (even if the optimal
stopping time for some types is not unique). The intuition is that the lower types get the
same payo¤ from selling as type c; but forego less of future cash‡ows. This is known as
the “skimming property” and it simpli…es equilibrium analysis since in equilibrium the set
of seller types remaining in the market at any time is a truncation of the original seller
distribution.
Let x (c) be some selection of the optimal stopping times given the net-price process,
pt (1
t) :
7
Let kt denote the lowest quality asset that has not been traded by time t :
kt = inf fc : x (c)
5
tg
Assuming v (1) = 1 allows us not to worry about out-of-equilibrium beliefs after a history where all
sellers were supposed to trade but some did not trade. The equilibria we characterize in this paper continue
to exist even if v (1) > 1 but may no longer be unique.
6
As discussed in Remark 2, the exact form of the subsidy turns out not to matter.
7
To assure the stopping problem has a solution, we restrict t to be such that we can construct equilibrium
pt so that pt (1
t ) is right-continuous when it is increasing and left-continuous when it is decreasing.
7
Note that kt is left-continuous and it is independent of the selection of the optimal stopping
times (since for any t at most zero measure of types are indi¤erent between stopping at that
time and some other time).
We use Kt to describe the (set of) types that trade at t: There are three possibilities: (i) if
kt is constant to the right of t; that means there is no trade at t; and we denote it by Kt = ;;
(ii) if kt increases continuously to the right of t; it means trade is smooth at t; and we denote
it by Kt = kt ; (iii) if kt jumps discontinuously from kt to kt+ = lims!t+ ks ; it means that
there is an atom of trade at t; and we denote by Kt = [kt ; kt+ ] :8
With this notation we de…ne a competitive equilibrium:
De…nition 1 Given a tax schedule f t g ; a competitive equilibrium is a pair of functions
fpt ; kt g for t
0 that satisfy:
(E1) Zero Pro…t Condition: if Kt 6= ;; then pt = E [v (c) jc 2 Kt ]
(E2) Seller Optimality: Sellers optimally choose their stopping times (i.e., kt is consistent
with seller optimization given pt and
(E3) Market Clearing: for all t; pt
t ).
v (kt ).
Conditions (E1) and (E2) are standard. Condition (E3) deserves a bit of explanation.
It is needed because condition (E1) provides no discipline when Kt = ;: We justify it by
a market clearing reasoning, that is, that given the market prices demand equals supply.
Suppose at some t the assets were o¤ered at pt < v (kt ) : Then, since all buyers believe that
the value of the asset is at least v (kt ) ; they would all demand it. Demand would not equal
supply and the market would not clear.9 This condition removes some trivial multiplicity of
equilibria. For example, it removes as a candidate equilibrium the path (pt ; kt ) = (0; 0) for
all periods (i.e. no trade and very low prices) even though this path satis…es the …rst two
conditions.10
We assume that all market participants publicly observe all the trades. Hence, once a
buyer purchases an asset, if he tries to put it back on the market, the market makes a
correct inference about c based on the history. Since we assume that all buyers have the
same value of the asset, there would not be any pro…table re-trading of the asset (after the
initial seller transacts) and hence we ignore that possibility.
8
We use the notation kt+ ; pt+ ; etc., to denote right-limits of the corresponding functions at t.
We thank Andrew Postlewaite for pointing this out to us.
10
Condition (E3) is analogous to the condition (iv) in Janssen and Roy (2002) and is weaker than the No
Unrealized Deals condition in Daley and Green (2012) (see De…nition 2.1 there; since they study the gap
case, they need a stronger condition to account for out-of-equilibrium beliefs).
9
8
2.1
Laissez-faire Equilibrium
Absent government interventions, i.e., if
t
= 0 for all t; the equilibrium has no atoms of
trade and is given by:
Proposition 1 (laissez-faire) If
t
= 0 for all t then there exists a unique competitive
equilibrium that is the unique solution to:
pt = v (kt )
k0 = 0
r (v (kt )
kt ) = v 0 (kt ) k_ t
(2)
To see the intuition (proof is in the appendix), note that if an interval of types traded
at time t; Condition (E3) would require that an instant later pt+
v (kt+ ) : However, that
would imply a jump in prices at t since pt = E [v (c) jc 2 Kt ] < v (kt+ ) : But if prices jumped,
no type would trade the instant before the jump. Hence, kt is a continuous function in
equilibrium. Therefore, by Condition (E1); pt = v (kt ) when there is trade. The di¤erential
equation for kt comes from seller optimality: at each point in time, the current cuto¤ type kt
must be indi¤erent between trading or delaying trade for an instant. The gain from waiting
for an instant of time is that prices rise over time while the cost of delaying trade is the lost
interest on the gains from trade. Together:
r (pt
kt ) = p_t
Using pt = v (kt ) ; this tradeo¤ can be stated as:
r (v (kt )
kt ) = v 0 (kt ) k_ t
(3)
This di¤erential equation, together with the boundary condition k0 = 0; pins down the
equilibrium path of kt .
Note that if v (0) = 0 (i.e. no strict gains from trade in the bottom of the distribution),
there is no trade in equilibrium.11 Moreover, the dynamics of (pt ; kt ) in the laissez-faire
equilibrium do not depend on the shape of the distribution F (c) but only on its support.
The shape of F (c) will play a role once we introduce government interventions that generate
11
This can be seen from equation (3) : Continuous trading is not necessary for this result to hold. If trading
is at discrete intervals of time this result would continue to hold for small if f (0) > 0: Even for = 1
the result may continue to hold, as we show in Section 4.3.
9
atoms of trade. Finally, total surplus/gains from trade in the laissez-faire equilibrium are:
SLF =
Z
1
e
rt
kt ) k_ t f (kt ) dt =
(v (kt )
Z
1
e
rt~(c)
(v (c)
c) f (c) dc
0
0
where t~(c) is the inverse of kt . Condition (3) implies t~0 (c) =
v 0 (c)
r(v(c) c)
and hence SLF is
independent of the discount factor (since c and v (c) are present values, the …rst-best surplus
is independent of r as well). The intuition is that since in equilibrium all types eventually
trade, the deadweight loss is due to the delay of trade. While a smaller discount implies
that any …xed delay is less costly, by condition (3) a smaller r implies more delay. Since the
speed of trading, k_ t ; is proportional to r; these two forces cancel each other out.
3
Government Interventions: Motivating Examples
We start with the following benchmark example to illustrate the bene…ts of imposing future
taxes to increase early trading. Assume c is distributed uniformly over [0; 1] and v (c) =
as illustrated in Figure 1.
1.0
v(c)
0.5
sf
n
gai
de
rom
tra
c
0.4
0.6
0.0
0.0
0.2
0.8
1.0
Figure 1: Example of gains
from trade.
Laissez-faire Economy
Absent any government interventions in this example the equilibrium cuto¤s are:
ktLF = 1
e
rt
:
The total surplus in the laissez-faire economy is:
SLF =
Z
1
e
rt~(c)
(v (c)
0
10
1
c) f (c) dc = :
6
1+c
;
2
Note that even though asymptotically all types trade in equilibrium, the equilibrium is
ine¢ cient due to delay. The …rst-best has all types trading immediately (i.e. t~(c) = 0) and
R1
surplus is 0 (v (c) c) f (c) dc = 14 > SLF :
Initial Subsidy Followed by Constant Permanent Tax
Now consider the following government intervention. The government provides an initial
subsidy s =
tax rate
t
0
=
0 per unit traded at time 0 and then …nances this subsidy with a constant
for t > 0; to (dynamically) achieve budget balance. One interpretation of
this intervention is that the government levies a constant tax rate on the market to …nance
a subsidized auction at time 0:
To construct the equilibrium we solve the following …xed point problem. We …rst solve for
the equilibrium for any s and : Then, we look for pairs of (s; ) that are budget neutral.
The amount of initial trade depends on the initial subsidy; how much of a subsidy can be
provided depends on the amount of trade after t > 0 which, in turn, is a function of which
types trade at t = 0:
The one-time subsidy at time 0 induces an atom of trade at time 0 and the constant tax
later implies either no trade (if the tax is high enough) or smooth trade: The equilibrium
conditions are as follows. First, cuto¤ type at time 0;
k0+ ; must be indi¤erent between
pooling with lower types and getting the initial subsidized price p0 ; or waiting an instant to
separate from them and getting a higher price but being taxed:
8
>
<
p0 (1 + s) = max v ( ) (1
>
:| {z }
);
=p0+
9
>
=
>
;
:
(4)
where the maximization captures the two possibilities: there will either be trade after t = 0
(with buyers paying p0+ = v ( )) or no trade at all.
Second, the buyer zero pro…t condition at time t = 0 is:
p0 = E [v (c) jc 2 [0; ]] :
(5)
The unique solution to (4) and (5) pins down the equilibrium
and price p0 given (s; ) :
Third, if there is any more trade after time t = 0 (i.e. if
is not too high) it must
be smooth. The same reasoning as in the laissez-faire equilibrium holds after time zero.
11
Equilibrium is then pinned down by the sellers’indi¤erence condition:
r (pt (1
)
kt ) = (1
) p_t
and the zero-pro…t condition pt = v (kt ) : Using the assumed form of v (c) and given a
boundary condition
, the unique solution of this di¤erential equation is:
kt =
1
1+
1
1+
e
r 11+ t
:
Inverting it, we get the following expression for the time at which each cuto¤ type k trades:
1
1
t~(k) =
r +1
ln
k (1 + ) +
(1 + ) +
1
1
Note that because of the tax, types such that (1
f or k 2
) v (c)
;
1
1+
:
c do not trade in equilibrium.
That completes the characterization of the equilibrium for any (s; ) :
Finally, to verify that in equilibrium the intervention is budget-neutral, we require that
for any …xed
the subsidy s satis…es:
sp0 =
Z
1
rt~(k) 1
e
+k
dk
2
(6)
The unique positive solution (s; ; p0 ) to (4) ; (5) ; (6) pins down the unique competitive
equilibrium in this example. With it we can then calculate the total surplus associated with
a given tax rate
:
S( )=
Z
(v (c)
c) dc +
0
Z
How does the equilibrium depend on ?
12
1
e
r t~(k)
(v (kt )
kt ) dk:
1.0
5%
0.8
15%
0.6
0.4
No tax
0.2
0.0
0
10
20
30
40
50
time
Figure 2: Taxes and Cuto¤ Dynamics
In Figure 2 we plot equilibrium cuto¤s, kt ; for di¤erent tax rates: the dashed line has
= 15%, the solid line
= 5% and the dotted line has
= 0%: As shown, a higher tax rate
leads to a higher initial cuto¤ but slower trade thereafter. In other words, as taxes increase,
there is a tradeo¤ between trading faster with the lower types at the expense of slower trade
with higher types.
How does the total surplus change with ?
1.0
0.9
Taxes used to pay subsidy at t=0
0.8
No initial subsidy
0.7
0.00
0.05
0.10
0.15
0.20
Tax rate
Figure 3: Surplus relative to …rst best
with a tax-exempt auction followed by a
constant tax.
Figure 3 shows that in this example surplus (the solid line, represented as a fraction of
…rst-best surplus) monotonically increases in the level of taxes. For su¢ ciently high taxes
(
20%); there is no trade after t = 0 and the surplus is only 11% below …rst best, With
no intervention it is 33% below.
13
Somewhat surprisingly, even if the taxes are levied but not used for the initial subsidy
(i.e. if s = 0 but
> 0); the surplus is also increasing in ; as shown by the dashed line
in Figure 3. Comparing the two curves, for small tax rates, the initial subsidy has a large
contribution to the welfare gains; yet, for large tax rates the di¤erence is small. The reason
is that for high tax rates the La¤er Curve implies that total tax revenues decrease in the
tax rate and hence the subsidy becomes again small. The main e¤ect of taxes is then that
they push more sellers to participate in the initial tax-exempt auction, and that improves
e¢ ciency.
4
Optimal Government Interventions
In this section we return to our general model and provide two results. First, we show that for
general F (c) and v (c) there is always room for the government to improve over laissez-faire
and even more, that a tax-exempt auction followed by a short-lived tax achieves a Pareto
improvement. This improvement is possible even if the government can only use budgetneutral polices. Then, under a regularity condition on F (c) and v (c) ; we characterize the
optimal interventions for a zero (and positive) budget constraint.
4.1
Laissez Faire is Never Optimal
Our main result follows. Consider the following tax policy:
t
=
(
0 f or t 2 ff0g [ [ ; 1)g
>0
otherwise
That is, there is tax-exempt trading at t = 0; followed by a (short) time interval
in
which transactions are taxed at , after which taxes are reduced back to zero.
We show that for small
t
> 0 all sellers prefer
over the laissez-faire equilibrium (i.e.
= 0):
Theorem 1 Suppose v (0) > 0: For every r;
> 0; F (c) ; and v (c) ; there exists
> 0 such
that an equilibrium with government policy
yields strictly higher gains from trade than the
laissez-faire equilibrium. Moreover, it is preferred by all seller types (Pareto improvement).
To establish that the
intervention increases overall e¢ ciency we show in the proof (see
appendix for details) an even stronger result: for small
14
; under
every type trades sooner
than under laissez-faire. We start with characterizing the equilibrium with
for small
:
First, since after the initial period taxes increase discontinuously, it will attract an atom of
types to trade at t = 0: Let
be the highest type that trades at t = 0 with
the fact that taxes drop discontinuously at t =
: Moreover,
implies that for small
there will be
no trade in the time interval [0; ] (roughly, waiting increases prices from (1
v (k ) and for small
Let k
LF
) v (k ) to
that is a much larger bene…t than the discounting cost).
denote the equilibrium cuto¤ at time t =
show in the proof, for small
; k LF <
: Since with
with no taxes (laissez-faire). As we
after time
the equilibrium is the
same as in case of the laissez-faire economy but with a di¤erent boundary condition, every
type trades sooner under
and the claim follows.
The key step of the proof is to show that:
@
!0 @
lim
Since as
! 0 both
@k LF
:
!0 @
= 2 lim
and k LF converge to 0; this means that for small
twice as many types trade before
approximately
if the government intervenes in (0; ) : The intuition
is as follows. As we announce the tax plan
, some types that were planning to trade in
(0; ) now would prefer to trade at 0 even if the price at 0 did not change: The reason is
that not taking the price p0 implies a …xed delay cost. It turns out that the set of types that
decide to take that …xed p0 grows in
approximately as fast as does k LF .
The doubling of early trade is then achieved because pooling of trade at time 0 reduces
adverse selection faced by buyers and hence price p0 increases. For small
approximately half way between v (0) and v (
the price is
). As the price goes up, even more types
prefer to trade at 0 and the adverse selection problem is reduced even further, making p0
even higher, and so on. Because prices grow at half the speed of v k LF , the resulting cuto¤,
;
is twice as high as k LF :
To see the improvement is Pareto, note that type
p = v(
could choose to trade at t =
for
) which is strictly better than what he would get in the laissez-faire economy since
it would take the economy longer to reach that price absent taxes (and in equilibrium he
trades at that price). Types c >
are also better o¤ because they trade at the same price
but earlier than they would absent the intervention. To see that types c <
are better o¤,
note that given that they all trade immediately their payo¤ ends up being the same as that
for c =
whereas in the laissez-faire economy it would be strictly lower.
The same logic can be used to show that even when initial trades are subsidized (i.e.
0
< 0 even if this subsidy is not …nanced by subsequent taxes), it is still optimal to introduce
15
positive taxes into the market for some small time
:
We required for the result that v (0) > 0: If v (0) = 0 whether the tax policy
leads to
a strict increase of surplus or no change depends on whether it induces any trade. That, in
turn, depends on the shape of F (c) and v (c) : Both cases are possible: in the example in
p
Section 4.3
does not induce trade for any , while in case F (c) = c and v (c) = c; if
e
< 23 ; then
r
4.2
does induce trade and thus generates a Pareto improvement.
When Extreme Policy is Optimal
We showed so far that a short-lived budget-neutral intervention improves over the laissezfaire equilibrium and that in fact we can get a Pareto improvement. Although the proof of
Theorem 1 uses
for
! 0 it is important to note that the optimal policy would generally call
>> 0: Furthermore, the intervention can also continue to be a Pareto improvement
outside of the limit. If a small intervention generates an improvement, what a about a larger
one? For our example economy, with c distributed uniformly over [0; 1], v (c) =
r = 10%; we can see in Figure 4A that total welfare is actually increasing in
we show the surplus gain/loss of the cuto¤ type,
1+c
2
, and
: In Figure 4B
; over what his surplus would be without
12
any intervention.
0.03
0.8
0.02
0.01
0.7
2
4
6
8
Winners & Lossers
∆
0.00
∆
0
Pareto
Improvement
2
4
6
8
10
-0.01
10
Figure 4A: Total surplus 4B: Gain/loss of cuto¤ type
relative to …rst best.
relative to no intervention.
Is this true in general? To answer this question we introduce the following condition:
De…nition 2 We say that the environment is (strictly) regular if
f (c)
F (c)
(v (c)
decreasing.
12
The intervention is a Pareto improvement if the cuto¤ type is weakly better o¤.
16
c) is (strictly)
As we explain below, the ratio
f (c)
F (c)
c) represents the relative marginal e¤ect of
(v (c)
speeding up trade of type c on the social surplus and the information rent of the seller.13
Under this condition we can show that the optimal policy induces trade that is extremely
concentrated in time:
Theorem 2 If the environment is regular, a competitive equilibrium for a tax policy
t
=
(
0 for t = 0
H otherwise
where H is high enough so that there is no trade after t = 0; maximizes total surplus over
all possible budget-neutral policies and all corresponding equilibria.
Moreover, if the environment is strictly regular, the competitive equilibrium for this tax policy
is unique.
The benchmark example in Section 3 (v (c) =
1+c
2
and F (c) = c) satis…es the regularity
condition and hence we have described there the optimal budget-neutral intervention: taxexempt initial trade followed by prohibitively high taxes for t > 0: Note that it is welfaremaximizing, but compared to the laissez-faire equilibrium there are winners and losers from
this intervention: for example, type 3/4 eventually trades in the laissez-faire economy, but
does not trade under this optimal intervention, so that type prefers the former. Therefore, it
would be harder to build consensus to implement these type of policies ex-post, once agents
know their types.
Our proof of Theorem 2 considers a mechanism design problem with a market designer
who maximizes expected gains from trade. The designer is allowed to cross-subsidize sellers
trading in di¤erent periods but has to break even on average.
Letting Gt (c) denote, for a given type, the (cumulative) distribution over times of trade,
the expected discounted time to trade for this type is:
x (c) =
Z
1
e
rt
dGt (c) ;
0
and since all the traders are risk-neutral, their expected payo¤s depend on Gt (c) only via
x (c) :
13
This regularity condition is also similar to the standard condition in price theory that the marginal
revenue is monotone. In particular, think about a static problem of a monopsonist buyer choosing volume of trade, F (c) ; by making a take-it-or-leave-it o¤er equal to P (c) = c: The FOC of this problem
is: f (c) (v (c) c) F (c) = 0: A decreasing ratio Ff (c)
c) guarantees that the marginal pro…t (the
(c) (v (c)
left-hand-side of the FOC) crosses zero exactly once.
17
In the direct revelation mechanism the designer chooses x (c) and a net transfer to type c;
P (c) ; to maximize:
max
x(c);P (c)
Z
1
x (c) (v (c)
(7)
c) f (c) dc
0
subject to budget-neutrality:
Z
1
[x (c) v (c)
P (c)] f (c) dc
0;
0
and truth-telling constraint:
c 2 arg max (1
x (~
c)) c + P (~
c)
c~
and individual rationality for the seller. Truth-telling implies that the equilibrium payo¤
of type c; U (c) ; has to satisfy U 0 (c) = (1
x (c)) almost everywhere: We use this to express
the budget constraint in terms of the allocation only, x (c) :
Z
1
(x (c) (v (c)
c)) f (c) dc
0
Z
1
x (c) F (c) dc
0
(8)
0
The …rst term is the amount of money the mechanism designer can collect from the buyers
and the second term is the information rent he has to pay the sellers so that they report
their types truthfully.
The ratio of derivatives of the constraint (8) and the objective function (7) with respect
to x (c) is
1
F (c)
(v(c) c)f (c)
which is decreasing under our regularity condition. That is a
su¢ cient condition for the optimal solution to have a bang-bang property: types below a
threshold trade immediately and types above the threshold never trade. That solution can
be implemented by a competitive equilibrium imposing su¢ ciently high taxes after the initial
trade. Given these taxes, if the regularity condition holds strictly, the equilibrium is unique.
Hence this extreme intervention is the most e¢ cient.14
Remark 1 Without the regularity condition, it can be shown that the optimal policy (with
zero or positive-balance interventions) induces trade in at most two periods, with t = 0 being
14
Details of the proof are in the appendix. The proof uses standard mechanism design tools, similar
to Samuelson (1984) in a static environment. The contribution of our proof is to apply these methods to
characterize optimal intervention in a dynamic market. On the technical side, we contribute by establishing
a su¢ cient condition for the optimal mechanism having trade only at t = 0; and showing that the described
policy induces a unique competitive equilibrium that implements the outcome of the optimal direct revelation
mechanism.
18
one of them. Thus, except at these two times, the government sets su¢ ciently high taxes so
that no trade would take place.
Our two main results are in stark contrast to recent results in the literature. Optimal
government interventions in similar models (although, admittedly richer) have been studied
recently by Philippon and Skreta (2012) and Tirole (2012). In these papers, the government
o¤ers …nancing to …rms having an investment opportunity and it is secured by assets that the
…rms have private information about. That intervention is followed by a static competitive
market in which …rms that did not receive funds from the government can trade in a private
market to raise funds. This creates a problem of "mechanism design with a competitive
fringe" as named by Philippon and Skreta (2012): the government intervention a¤ects the
post-intervention equilibrium and vice versa. This e¤ect is shared by our model: for example,
under the policy in Theorem 1, the amount of trade at t = 0 under
depends on the price
after the taxes are removed and that in turn depends on which types trade at time 0:
Both papers obtain a result that shutting down the private market does not improve welfare: see Proposition 2 in Tirole (2012) and Theorem 2 in Philippon and Skreta (2012). Since
the post-intervention market creates endogenous IR constraints for the agents participating
in the government program, making the market less attractive could make it easier for the
government to intervene. However, these two papers argue that this is never a good idea.
As hinted by our benchmark example and generalized in Theorems 1 and 2, taking into
account the dynamic nature of the market changes this conclusion. The key di¤erence in
the models that leads to these opposing results is that both Philippon and Skreta (2012)
and Tirole (2012) assume a static model of the private market, while we study a dynamic
market. In our example setting tax rate prohibitively high so that all trade takes place
at t = 0 turns out to be equivalent to assuming that the private market is opened only
once after the government intervention, as in their papers. Theorem 1 showed that some
shut-down is always optimal and Theorem 2 showed that under our regularity condition a
complete shutdown is optimal within our model.
We don’t think one should take this extreme policy as a literal recommendation of how
governments should intervene since this extreme policy is unlikely to be optimal in a richer
environment where additional shocks or gains from trade continue to arise over time. An
additional consideration with closing the market forever is that it might require a lot of
commitment power from the government - a shorter intervention would su¤er less from such
a concern. Therefore, we think that Theorem 1 is a more realistic policy prescription.
19
4.3
Non Budget-Neutral Interventions: Jump Starting the Market
We now show via an example that if the adverse selection problem is su¢ ciently severe,
even the best budget-neutral intervention is incapable of generating any trade. Consider the
following example: F (c) is uniform in [0; 1] and
(
v (c) =
1:5c if c <
1+c
2
if c
1
2
1
2
This example satis…es the regularity condition. Hence, Theorem 2 implies that the optimal
budget-neutral intervention would induce trade only at t = 0: Unfortunately, as in Akerlof’s
original example, even if there is only one opportunity to trade, we get complete unraveling.15
This follows since the equilibrium cuto¤ type
must satisfy:
= p0
but the zero-pro…t condition is
p0 = E [v (c) jc < ] :
Since in this example E [v (c) jc < ]
3
4
; the only solution is
Now suppose the government allocates a budget b
= 0:
0 to bailout this market. Suppose it
uses it to subsidize the initial trades with a proportional subsidy s: When b <
1
16
the optimal
p
intervention has s = (and prohibitively high taxes after t = 0); types below 0 = 2 b
p
trade at t = 0 and buyers pay p0 = 23 b per unit. Total gains from trade (net of transfers)
1
3
are:
S (b) =
Z
p
2 b
(0:5c) dc = b
0
Since the buyers still break-even, every dollar the government spends increases the welfare
of the sellers by 2 dollars, one from the direct transfer and one from the improvement in
the e¢ ciency of the market (if b >
1
;
16
the marginal e¤ect is even higher). Thus, if the
deadweight loss associated with raising taxes from other markets is not too large, bailouts
are welfare-improving.
A natural question to ask is if b > 0; and so the government can restart the market at
15
The classic example from Akerlof has v (c) = 1:5c and no trade in (static) equilibrium. We keep close to
his example, but modify v (c) slightly to have v (1) = 1: Note that it implies even less gains from trade than
when v (c) = 1:5c.
20
t = 0; would it be even better to have low taxes after t > 0 so that there would be additional
trade (and government could use the revenues to …nance an even larger subsidy at t = 0)?
Theorem 2 describes the optimal zero-budget policy, but the proof can be easily modi…ed to
allow the government a total budget b to be spent over time (including interest on savings)
by putting b on the right-hand side of (8) : This leads to:
Corollary 1 Suppose the government has a budget b
Then, a welfare-maximizing intervention has
0
=s
0 and the environment is regular.
0 and
t
H for t > 0; where H is
high enough that there is no trade after t = 0:
In other words, to maximize welfare given a budget b, all the subsidy should be used at
time zero and it would not be worth trying to raise additional revenue by taxing future trade
- instead, the future taxes should be high enough to induce as many types as possible to
trade at time 0.16
Remark 2 It is important to note that the exact form of the intervention is not important, as
long as the information rents collected by the sellers are unchanged. A proportional subsidy,
as assumed for concreteness in the paper, or a government guarantee on the payo¤ of the
assets, as implemented during the crisis with the Public-Private Investment Program for
Legacy Assets would have equivalent e¤ects. The outright purchase of assets done with the
TARP program is also equivalent as long as we take into account the proceeds in the budgeting
for the program, that is, consider just the expected net cost of the program as the government’s
budget.17
5
The Cost of Delaying Interventions
In practice it might take time for the government to act upon a crisis. In this section we
show that speed is often of the essence. Not only is it usually optimal to act immediately, as
established by Theorem 2, but delayed interventions can actually decrease surplus compared
to the laissez-faire equilibrium. In particular, we show that if the government is expected
to provide a bailout at some future time, it slows down (or even shuts down) trade before.
Even though the bailout has a positive direct e¤ect on e¢ ciency, it creates also this negative
16
If b is large enough, b 1 E [v (c)] ; …rst-best is achievable.
As we discussed in the Introduction, the government proceeds can either come from holding the assets
to maturity or alternatively from creating a portfolio and selling its shares.
17
21
endogenous/equilibrium e¤ect of delay due to anticipation. The net e¤ect can be negative:
the equilibrium welfare with a delayed subsidy can be lower than absent any intervention.
We show this claim via two examples using the benchmark example from Section 3: v (c) =
1+c
2
and F (c) = c. We consider …rst the case of a deterministic date for the intervention and
then the case when the timing of the intervention is uncertain.
5.1
Delayed Interventions at Announced Date
Consider the following policy:
T
=
(
s for T > 0
0 otherwise
The government has a dynamic budget constraint e
rT
spT jKT j
b; where jKT j is the
measure of types that trade at T and pT is the market price buyers pay at T; so that the
left hand side is the time-zero present value of the total subsidy at T:
For any b > 0 there exists a T such that if T
T there is no trade in equilibrium until
T since all sellers prefer to wait for the subsidy than to trade immediately. In this range of
T the competitive equilibrium has atom of trade at T followed by smooth trading as in the
laissez-faire equilibrium. The following conditions pin down the equilibrium for T
First, denote by
T .18
the highest type trading at T: This type has to be indi¤erent between
trading at the subsidized price and trading after the subsidy is removed:
pT (1 + s) = pT + = v ( ) ;
where the second equality follows because after T the equilibrium coincides with the laissezfaire equilibrium with a boundary condition kT =
(and hence the zero-pro…t condition
with smooth trading implies pt = v (kt )): The zero-pro…t condition for prices at T is:
pT = E [v (c) jc
]:
Finally, the budget constraint is:
e
rT
pT s = b:
18
When T is large, the analysis is more complex. The equilibrium then has continuous trading until some
time T ; from T to T; the market shuts down waiting for the subsidy, an atom of sellers trade at T; and
smooth trading follows from then on.
22
From these equations (using v (c) and F (c) from the benchmark example) we obtain:
p
= 2 beT r :
Assuming that b is small enough that
< 1; after T trade is smooth and we can use the
characterization of the laissez-faire equilibrium in Section 3 to compute:
kt = 1
(1
Inverting it yields the time at which types k >
t~(k) =
1
r
r(t T )
)e
ln
:
trade:
1
1
k
+ T:
That completes the characterization of the equilibrium.19
Present value of the gains from trade in equilibrium is:
S b; T < T = e
rT
Z
0
1
c
2
dc +
Z
1
1
1
c
1
c
2
dc
Since b = 0 corresponds to no intervention, we have S (0; T ) = SLF = 16 :
How does delay, T; a¤ect gains from trade?20 There are two opposing e¤ects. On one
hand, later subsidy implies that there is more delay until the market starts trading at T:
On the other hand, since the unused budget earns interest, it allows for more e¢ cient trade
at T and afterwards. It turns out that in our benchmark case the …rst force is stronger. In
p
particular, for b > 0 and T T and 2 beT r < 1 we get the following result:
Proposition 2 Assume v (c) =
1+c
2
and F (c) = c:
1) Despite the government being able to save at rate r; the equilibrium welfare is decreasing
in T;
@S(b;T )
@T
< 0.
2) Moreover, there exists T < T such that the subsidy delayed by more than T destroys
surplus, that is S (b; T ) < SLF for T 2 T ; T :
In words, delay is costly despite the budget growing with delay. Even more surprisingly,
the second part of the proposition states that the decrease in the surplus can be so large as
2
It implies T is the solution to e rT (1 + s) pT = v (0) which simpli…es to a solution of b = 41 (1 zz) where
z = e rT
20
The e¤ect of b is obvious: if b is small enough so that not all types trade at T; S (b; T ) is increasing in b
for all T; because higher b uniformly speeds up trade.
19
23
to drive the total surplus below the surplus with no intervention. For example with b =
and r = 1; if T > T = 0:387 then S (b; T ) <
1
6
1
10
= SLF (in this case T = 0:622 ): This is
illustrated in Figure 5.
Surplus from Delayed Intervention
S(0.1,T)
0.20
Laissez-faire
0.15
0.0
0.1
0.2
0.3
0.4
0.5
T
Figure 5: The Cost of Delayed Interventions.
5.2
Delayed Interventions with Uncertain Timing
In practice the market might expect the possibility of a government subsidy but be uncertain about its timing. We argue that this creates incentives to wait for the arrival of the
intervention and may be detrimental to welfare even taking into account the bene…ts of the
subsidy if it materializes.
To illustrate this problem, we analyze a model in which the government intervention
arrives at a random time as a Poisson process with intensity : Suppose that when it …nally
intervenes, the government subsidizes trade su¢ ciently that all types trade (by o¤ering
su¢ cient subsidy for pt (1 + s) = 1):
The equilibrium dynamics depend crucially on the level of : If
is small, there will be
trade even before the government subsidy arrives. If it is high, the market will shut down
completely until the arrival of the subsidy.
We start with the …rst, more interesting case. If
< r; then trade is smooth until arrival
of the subsidy. Equilibrium cuto¤s kt are characterized by the following di¤erential equation
(with a boundary condition k0 = 0):
r (v (kt )
kt ) = v 0 (kt ) k_ t + (1
v (kt ))
(9)
The left-hand side is the familiar discounting cost of not taking the price today. The …rst
term on the right-hand side is the familiar increase in price in case the intervention does not
24
arrive (as before, both use the zero-pro…t condition pt = v (kt )): The new term is the last
term on the right-hand side: by delaying trade, the current cuto¤ type can hope to receive
the subsidized price 1 instead of the non-subsidized price v (kt ) :
A higher
has two opposing welfare e¤ects. The direct e¤ect is that it speeds up the
arrival of the subsidy, which increases welfare. The indirect, equilibrium e¤ect, is that it
slows down trade before arrival and it decreases welfare (the indirect e¤ect can be seen from
(9) since k_ t that solves it is decreasing in ):
Connecting this observation to real-life events, market participants’ beliefs that Federal
Reserve and/or the US Treasury would intervene in some …nancial markets, might have
contributed to the reduction of trade volume in some of the those markets after the collapse
of Lehman Brothers. While asymmetric information was likely the primary culprit, the
expectations of future actions could make it worse.
Using v (c) and F (c) from the benchmark example, the di¤erential equation (9) simpli…es
to:
(r
) (1
kt ) = k_ t :
After solving it and computing total welfare, we can show that the negative, indirect e¤ect
always dominates (unless
is so high that the market shuts down completely):
1+c
2
Proposition 3 Assume v (c) =
and F (c) = c: Suppose the government subsidy that
induces …rst-best trade arrives at a Poisson rate :
1) If
< r; there is trade in equilibrium even before the subsidy arrives and the equilibrium
welfare is strictly decreasing in :
2) If
> r then the market shuts down in the anticipation of the subsidy and equilibrium
welfare is increasing in
(as
! 1 the surplus converges to …rst-best). In that range,
the surplus is higher than the laissez-faire equilibrium surplus if and only if
is su¢ ciently
higher than r.
In words, over a large range of arrival rates, the expectation of the possibility of arrival
reduces welfare in our benchmark example. In fact, for the delayed subsidy to have a hope
at improving welfare, the arrival rate has to be su¢ ciently high so that the market closes
down completely.
Figure 6 illustrates the results for r = 1:
25
Intervention with Random Arrival
0.18
0.16
Laissez-faire
0.14
0
1
2
3
Arrival Rate
Figure 6: Cost of Random Arrival
of Subsidy.
The intuition behind the second part of the above proposition is straightforward. For
su¢ ciently large there will not be any trade until the government intervenes. Hence, there
is only the …rst, direct e¤ect. Since equilibrium surplus is continuous in
from SLF as
and it decreases
increases from 0 to r; it has to increase su¢ ciently more to recover back to
SLF .
6
Conclusions
In this paper we have analyzed government interventions in a dynamic market with asymmetric information. Our main result is that even without the use of subsidies, e¢ ciency
can be improved over the laissez-faire equilibrium. Setting high taxes for a short period,
after an initial tax-exempt auction or trading window, induces more sellers to trade early
and enhances e¢ ciency. Remarkably, under a fairly commonly used regularity condition, the
optimal government policy is to set high taxes for t > 0; e¤ectively shutting down private
markets. Of course these results have to be interpreted with caution since they rely on the
assumption that the liquidity shock arrives once at the same time for all sellers. In reality,
since additional shocks may arrive over time, the optimal policy has to balance the ‡exibility
to respond to new shocks and the bene…ts of pooling trade early. Finding an optimal policy
in such an environment would be additionally complicated because, for the same reasons
as we discuss in Section 5, if current market participants expected future shocks and corresponding future interventions, it would create additional negative e¤ects of slowing down
trade.
Freeing the government from the requirement that its intervention must be budget-neutral,
26
i.e. allowing for bailouts, can be very valuable.21 Moreover, although the particular form in
which the subsidy is provided is not important, its timing is. Subsidies can greatly enhance
welfare when provided immediately or quickly after the shock, but they can even destroy
surplus if they are delayed.
It is natural to ask how these insights extend to normal times when we might still think
there is some amount of adverse selection in the market. Several interesting complications
can arise. For example, as opposed to the whole market being hit by a liquidity shock, as in
the recent …nancial crisis which gives us a clear notion of time zero, in many markets the time
the game actually starts is ill-de…ned and/or sellers arrive to the market at di¤erent times.
Janssen and Karamychev (2002) show that equilibria in dynamic markets with dynamic
entry can be qualitatively di¤erent from markets with one-time entry if the "time on the
market" is not observed by the market (see also Hendel, Lizzeri and Siniscalchi 2005 and Kim
2012 about the role of observability of past transaction/time on the market). As pointed out
recently by Roy (2012), a dynamic market can su¤er from an additional ine¢ ciency if buyers
are heterogeneous, because the high valuation buyers are more eager to trade sooner and
it may be that they are the e¢ cient buyers of the high quality goods. Incorporating these
considerations into our design questions would introduce new tradeo¤s which are interesting
avenues for future research.
7
Appendix
Proof of Proposition 1.
First note that our requirement pt
v (kt ) implies that there
cannot be any atoms of trade, i.e. that kt has to be continuous. Suppose not, that at time
s types [ks ; ks+ ] trade with ks < ks+ . Then at time s + " the price would be at least v (ks )
while at s the price would be strictly smaller to satisfy the zero-pro…t condition (1E). If so,
then for small " all types in [ks ; ks+ ] would be better o¤ not trading at s; a contradiction.
Therefore we are left with processes such that kt is continuous and pt = v (kt ) at any time
such that k_ t+ > 0: If kt is strictly increasing over time, we need that r (pt kt ) = p_t : if price
was rising faster, current cuto¤s would like to wait, a contradiction. If prices were rising
slower, over any time interval starting at s, there would be an atom of types trading at s,
another contradiction. So the only remaining possibility is that kt is constant over some
interval [s1 ; s2 ] : Since the price at s1 is v (ks1 ) and the price at s2 is v (ks2 ) ; if there is indeed
no trade in that time interval, then ps1 = ps2 : But then there exist a positive measure of
21
Note that these bailouts do not a¤ect the ex-ante incentives of the banks to screen the quality of their
assets but they do a¤ect their choices of liquidity. Potentially inducing them to take illiquid positions.
27
types k > ks1 such that
er(s2
v (ks1 ) > 1
s1 )
k + er(s2
s1 )
v (ks1 )
Since after s2 there are no atoms of trade, the equilibrium continuation payo¤ of types
er(s2
k > ks1 is smaller than 1
k+er(s2
s1 )
s1 )
v (k) since these types trade at price v (k) but
later than t = s2 : Since v (k) is continuous, there exists an " such that types k 2 [ks1 ; ks1 + "]
would strictly prefer to trade at t = s1 than to follow the postulated equilibrium. That leads
to the …nal contradiction.
Proof of Theorem 1. To establish that the market with
is more e¢ cient than under
laissez-faire, we show an even stronger result: that for small
there is more trade at t = 0
with
than with
t
= 0 over the time interval [0; ] : Since under
the equilibrium after
is characterized by the same di¤erential equation as under laissez-faire, if the boundary
condition at
Let
is higher, all types trade faster in equilibrium under
be the highest type that trades at t = 0 under
equilibrium cuto¤ at time
under laissez-faire: Since lim
see discussion in Step 1 below), to establish that
show:
@
!0 @
lim
Step 1: Characterizing lim
Consider policy
@
!0 @
>k
= k0+ ): Let k LF the
(i.e.
!0
LF
:
= lim
for small
!0
k LF = 0 (for
; it is su¢ cient to
@k LF
!0 @
> lim
:
:
First notice that since
t
> 0 for t 2 (0; ] and
=
t
= 0 for t >
; for small
there
cannot be any trade in t 2 (0; ]: Suppose not. Let k be the supremum over types that
trade in that time interval: All types that trade in that interval get a payo¤ no higher than
) v (k ) (buyers would lose money if they paid more than p = v (k ) ; the government
(1
takes
of that price, and the best case scenario is that they trade with no delay). For small
; that payo¤ is smaller than (1
e
r
)k + e
r
v (k ) : Since p
+
v (k ) ; all types that
trade in (0; ] would be strictly better o¤ waiting for the tax to be removed, a contradiction.
When the taxes are removed after t =
; the continuation equilibrium is unique and
is characterized in Proposition 1 albeit with a di¤erent starting lowest type. Namely, for
t>
:
pt = v (kt )
r (v (kt )
kt ) = v 0 (kt ) k_ t
28
with a boundary condition:
k =
:
The break even condition for buyers (1E) at t = 0 implies:
p0 = E [v (c) jc 2 [0;
Seller optimality (2E) implies that type
at t = 0 and selling for p
+
v(
must be indi¤erent between trading at this price
) at t =
= v(
)
p0 = 1
:
e
r
(v (
Combining these two conditions we get that
v(
For small
E [v (c) jc 2 [0;
)
]] :
)
)
is a solution to:
]] = 1
e
r
(v (
)
)
(10)
this equation has a unique solution. Using implicit function theorem we can
show that:
@
!0 @
lim
Intuitively, for small
, E [v (c) jc
=
c 2 [0;
2rv (0)
v 0 (0)
]]
v(0)+v(
2
)
(because we have assumed
that f (c) and v (c) are positive and continuous). So the bene…t of waiting, the left-hand side
of (10) ; is approximately v(
) v(0)
;
2
while the cost of waiting, the right-hand side of (10) ; is
approximately r v (0) : So for small
;
v(
)
solves approximately
v (0)
2
which yields
@
@
=
2rv(0)
v 0 (0)
as
Step 2: Characterizing lim
= r v (0)
! 0:
!0
@kLF
@
:
Consider the laissez-faire economy. Since kt is de…ned by the di¤erential equation
r (v (kt )
for small
kt ) = v 0 (kt ) k_ t ;
:
k LF
r
29
v (0)
;
v 0 (0)
and more precisely:
@k LF
rv (0)
= 0
:
!0 @
v (0)
lim
Summing up steps 1 and 2, we have:
@
lim
!0 @
@k LF
= 2 lim
!0 @
which implies the claim.
Step 3: Pareto Improvement
Take any
all types c >
such that there is no trade in t 2 (0; ] under
and that k LF <
trade at the same price but sooner in the market with
: Since
than in the
laissez-faire economy, it is immediate that they all prefer the former.
Type c =
trade at p = v (
also strictly prefers
) at t =
: while he is trading at t = 0, he has the option to
while in the laissez-faire economy he trades at the same price
but later. By revealed preference he is strictly better o¤ under
Finally consider any type c <
U (c) = U (
:
: All these types get the same payo¤ under
) = E [v (c) jc 2 [0;
]] for all c
:
:
On the other hand, it is immediate that in the laissez-faire economy the equilibrium payo¤
is weakly increasing in type (by revealed preference, type c0 can trade at the same time and
price as any type c < c0 and since c0 gets a higher payo¤ ‡ow from holding his asset, his
payo¤ from this strategy is at least as high as type’s c): Combining these observations yields:
ULF (c)
ULF (
)<U (
) = U (c) for all c
and that …nishes the proof.
Proof of Theorem 2. We use mechanism design to establish the result. The mechanism
designer chooses a direct revelation mechanism that maps reports of the sellers to a probability distribution over times they trade and to transfers from the buyers to the mechanism
designer and from the designer to the sellers. The constraints on the mechanism are: incentive compatibility for the sellers (to report truthfully); individual rationality for the sellers
and buyers (sellers prefer to participate in the mechanism rather than hold the asset forever
and the buyers do not lose money on average); and that the mechanism designer does not
lose money on average.
Using the regularity condition, we characterize a direct mechanism that maximizes dis30
counted gains from trade. We then show that if the environment is regular, the postulated
policy has a corresponding equilibrium that implements the outcome of this best mechanism.
An optimal mechanism leaves the buyers with no surplus (since he could reduce the payment to the buyers and use the savings to increase e¢ ciency of trade). Hence, we can focus
on general direct revelation mechanisms described by 2 functions, x (c) and P (c) ; where x (c)
is the discounted probability of trade over all possible trading times and P (c) is the transfer
received by the seller. Letting Gt (c) denote for a given type the distribution function over
the times of trade:
x (c) =
Z
T
rt
e
dGt (c) :
0
Since all players are risk neutral, the mechanism depends on Gt only via x (c) :
The objective function of the mechanism designer is to maximize
max
x(c);P (c)
Z
x (c) (v (c)
c) f (c) dc:
(11)
We now describe the constraints.
The seller’s value function in the mechanism is:
U (c) = P (c) + (1
(12)
x (c)) c
= max
P (c0 ) + (1
0
c
x (c0 )) c
(13)
Using the envelope theorem:
U (c) = U (1)
Z
1
(1
x (c)) dc
(14)
c
Seller IR constraint is U (c)
c and in the optimal mechanism it binds at c = 1.22
Incentive compatibility for the sellers requires that the envelope formula (14) holds and that
x (c) is weakly decreasing.
Since the buyers are willing to pay at most
the seller is:
Z
1
(x (c) v (c)
R1
0
x (c) v (c) f (c) dc; the budget constraint of
P (c)) f (c) dc
0
0
22
Since U 0 (c) = 1 x (c)
1; if the IR constraint is satis…ed at c = 1; it is satis…ed for all types. IR
binds at c = 1 since otherwise the mechanism designer could reduce P (c) by a constant and still satisfy all
constraints.
31
From (12) ; we can write P (c) as:
P (c) = U (c)
(1
x (c)) c
Substituting this to the left-hand-side of the budget constraint we get express it as a
function of the allocation alone:
Z 1
(x (c) (v (c)
c)) f (c) dc
0
Z
1
(U (c)
c) f (c) dc
0
0
We can use (14) and integration by parts to write the last term as
Z
1
(U (c)
c) f (c) dc
0
(c) jc=1
c=0
= (U (c) c) F
Z 1
x (c) F (c) dc
=
Z
1
(U 0 (c)
1) F (c) dc
0
0
to obtain the …nal form of the budget constraint:
Z
1
(x (c) (v (c)
c)) f (c) dc
Z
1
x (c) F (c) dc
(15)
0
0
0
The …rst term of the constraint is the revenue the designer can obtain from the buyers
and the second term is the information rent he has to pay the sellers to participate in the
mechanism.
We now optimize (11) subject to (15) ; ignoring necessary monotonicity of x (c) that assures
that reporting c truthfully is incentive compatible (we check later that it is satis…ed in the
solution).
The derivative of the Lagrangian with respect to x (c) is:
L (c) = (v (c)
where
c) f (c) 1 +
1
F (c)
(v (c) c) f (c)
> 0 is the Lagrange multiplier.23
Note that L (c) is (weakly) positive for c = 0 (strictly if v (0) > 0): Suppose
f (c)
F (c)
(v (c)
c)
is decreasing (which is our regularity assumption). Then L (c) crosses zero only once because
23
is strictly positive in the solution since otherwise the budget constraint would not be binding and we
would get x (c) = 1 for all c (…rst-best), but that would violate 15.
32
1+
1
F (c)
(v(c) c)f (c)
is decreasing and (v (c)
solution to
1+
1
c) f (c) is positive. Let c be the largest
F (c)
(v (c) c) f (c)
= 0:
An optimal x (c) is then:
1 if c c
0 if c > c
x (c) =
Since x (c) is monotone, a mechanism with this allocation (and appropriate P (c)) is incentive compatible.
That describes the optimal allocation in the relaxed problem: there exists a c such that
types below c trade immediately and types above it never trade. The higher the c ; the
higher the gains from trade.
The largest c that satis…es the budget constraint (15) is the largest solution of:
E [v (c) jc
(16)
c ]=c
since the LHS is the IR constraint of the buyers and the RHS is the IR constraint of the c
seller.
A tax policy
0
= 0 and
t
= H for t > 0 clearly induces an equilibrium such that there is
trade only at time zero and that equilibrium satis…es (16) for some c . To …nish the proof,
we need to show that the solution to (16) exists and if the environment is strictly regular,
the solution is unique.
1) Existence. To see that there exists at least one solution to (16) note that
E [v (c) jc
k]
(17)
k
is continuous in k; positive at k = 0 and negative at k = 1: So there exists at least one
solution.
2) Uniqueness. To see that there is a unique solution under the regularity assumption,
note that the derivative of (17) at any k is
f (k)
(v (k)
F (k)
E [v (c) jc
33
k])
1
When we evaluate it at points where (16) holds, the derivative is
f (k)
(v (k)
F (k)
k)
1
and that is by assumption decreasing in k:
Suppose that there are at least two solutions and select two: the lowest kL and secondlowest kH : Since kL is the lowest solution, at that point the curve (17) must have a weakly
negative slope (since the curve crosses zero from above): However, our assumption implies
that curve has even strictly more negative slope at kH . That leads to a contradiction since by
assumption between [kL ; kH ] expression (17) is negative, so with this ranking of derivatives,
it cannot become 0 at kH :
rT
Proof of Proposition 2. Denote z = e
0
S (z) = z @
Z p 4b
z
0
1
c
2
Z
1
: Then the surplus can be written as:
1
0
1 c A
q
dc + p @
4b
4b
1
z
z
so the surplus is increasing in e
rT
rT
2
1
;1
2
2
1
dcA = z
6
1
1
b+ z
3
3
r
b
z
2
1 (1 z)
;
4
z
for T small enough that
< 1 (so
); we get
S b; T =
which is less than
c
:
Evaluating the surplus at T = T so that b =
that e
1
1
1
6
for all e
rT
1
4z
12z
< 1 (for z
z2
1
1 it is an increasing function and at z = 1 it is
1
):
6
Proof of Proposition 3.
As we explained in the text, in case r >
described by the di¤erential equation
(r
kt ) = k_ t :
) (1
With a boundary condition k0 = 0 it has a unique solution:
kt = 1
e
34
(r
)t
:
the equilibrium is
If the subsidy arrives at time t; total surplus is:
S (tj ) =
Z
t
e
rs
ks ) k_ s ds + e
(v (ks )
rt
1
(v (c)
c) dc
kt
0
=
Z
1 2 (r
4
) + re
3r 2
t(3r 2 )
Taking expectation over the arrival time:
S( )=
Z
1
S (tj )
t
e
dt =
0
1 2r
4 3r
which is decreasing in :
For the second part, when
> r; total surplus is
S( )=
1
4r+
since upon arrival the government induces the …rst-best surplus which is
1
4
in our benchmark
example.
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